For example, let's say you have an equation with planck's constant h, some mass m, and some velocity v. When you say "the characteristic length of the system is defined as h,m, and v to some powers that dimensionally result in a length," why does that length actually mean something and why isn't it just some random, unrelated length?
A particular example: In a lecture (video) I watched on quantum mechanics, the wave function of a quantum harmonic oscillator is being calculated.
At 29:40, he finds a characteristic energy and length of the system. He says the characteristic energy is pretty close to the ground state energy (double the actual value) and at 32:05 he says that the maximum displacement of the oscillator is probably about the characteristic length. Why does this work in general? It seems so arbitrary.
Also, how do you know what a characteristic quantity will represent in the system before doing any other calculations?
If it's relevant, I am good with calculus and differential equations, but I have only taken high school physics which didn't use calculus and every answer was always precisely calculated from formulas, we never did any characteristic quantities of systems.
Best Answer
In fact it does not really mean anything in the proper sense, but let us look at an example: take a simple pendulum of length $L$ and let us assume that, under the action gravity $g$, we want to calculate the period of its oscillations. The only parameters that can constitute an equation are, thus, $L$ and $g$ and the only combination thereof that results in a unit of time is $$ \sqrt{\frac{L}{g}} $$ as such, the period of a pendulum must be some coefficient times the above expression. Obviously, if you take the above per sé, it does not mean anything but eventually you find out that the proper expression is$^1$ $$ 2\pi \sqrt{\frac{L}{g}}. $$ Likewise, for more complicated matters, one can "cook" together the characteristic parameters appearing in the theory to have an "a priori" sense of what the proper scale may be: eventually, it will just be a matter of some numerical coefficients to multiply it by.
$^1$ The above was actually an entry question for general physics at the famous "Scuola Normale Superiore di Pisa" in Italy, where the candidate was asked to choose what the "most feasible" expression for the period of oscillation can be (had they not known the correct formula) between $ 2\pi \sqrt{\frac{g}{L}}$ and $\sqrt{\frac{L}{g}}$: the former has the correct coefficient $2\pi$ but swaps $L$ and $g$ (making it, thus, not a unit of time); the latter did not have the right coefficient, but still actually it could by all means be it, if you have to guess. The ones who chose the former were addressed in the answer as "not having the proper mindset for physics" (or something along those lines).